Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

270 ANALYTIC GEOMETRY
which states that R, in moving, stays always on the parabola.
The slope of the tangent at R is m/y', Ch. IX, ~ 2, eq. (5);
consequently, the line through 0 perpendicular to the tangent is
y'
y --- —-x
Em
As the equation of FR we have
Y= - ' ( 2m
2
The equations expressing the fact that P: (X, Y) is the
point of intersection of these two lines are, therefore,
(6)= - X,
m
(7) (- ) Y=- (- ).
From equations (5), (6), and (7) we have to eliminate x' and
y'. Solving (6) and (7) for x' and y', we have:, n( — m(-X)
X 2XY
Substituting these values for y' and x' in (5) and reducing the
result, we obtain
X2+ y2- mX = 0
as the equation of the locus.
The locus is therefore a circle, passing through the vertex of
the parabola and having its center at the focus. The vertex,
0, is not a point of the locus.*
Elimination of x', y'. In each of the above examples we
eliminated the auxiliary variables xa, y' by solving the last
two of a set of three equations for x', y' and substituting the
values thus obtained for x', y' in the first equation,-the
* It is an exceptional point, similar in type to the exceptional points,
A and A', of Example 1. For, when R is at 0, FR and OP coincide and
consequently determine no point on the locus.